A Theory of Network Tracing
H. B. Acharya1 and M. G. Gouda1,2
1
2
The University of Texas at Austin, USA
The National Science Foundation, USA
{acharya, gouda} @cs.utexas.edu
Abstract. Traceroute is a widely used program for computing the topology of any network in the Internet. Using Traceroute, one starts from a
node and chooses any other node in the network. Traceroute obtains the
sequence of nodes that occur between these two nodes, as specified by
the routing tables in these nodes. Each use of Traceroute in a network
produces a trace of nodes that constitute a simple path in this network.
In every trace that is produced by Traceroute, each node occurs either
by its unique identifier, or by the anonymous identifier“∗”. In this paper,
we introduce the first theory aimed at answering the following important
question. Is there an algorithm to compute the topology of a network N
from a trace set T that is produced by using Traceroute in network N ,
assuming that each edge in N occurs in at least one trace in T , and that
each node in N occurs by its unique identifier in at least one trace in
T ? We prove that the answer to this question is “No” if N is an even
ring or a general network. However, it is ”Yes” if N is a tree or an odd
ring. The answer is also “No” if N is mostly-regular, but “Yes” if N is a
mostly-regular even ring.
1
Introduction
Traceroute is arguably the most popular mechanism for computing the topology
of a network in the Internet [2] and [12]. Executing Traceroute between any two
nodes, say nodes x and y, in a network produces a sequence of node identifiers
that corresponds to a simple path between x and y in the network. This sequence
of node identifiers is usually referred to as a trace between x and y.
Traceroute can be used to compute the topology of a network N in the
Internet as follows [2] :
1. Identify the “terminal” nodes in network N (preferably at the perimeter of
N for good coverage).
2. Execute Traceroute between every pair of terminal nodes of N , identified in
Step 1, to produce traces of nodes that occur between each pair (as per the
routing tables in the nodes of N ).
3. Put the traces produced in Step 2 together in order to compute the topology
of network N .
It turns out that this procedure for using Traceroute to compute the topology
of network N has a problem. As observed in [13], [9], and [7], some of the
nodes in the traces produced in Step 2 occur by anonymous identifiers, rather
than by their unique identifiers. This causes Step 3 to compute many candidate
topologies, rather than one unique topology, for network N .
To solve this problem, Yao et al. [13] have suggested that Step 3 compute
only the topology with the smallest number of anonymous nodes, subject to
some constraints (trace preservation and distance preservation). This suggestion
has two problems of its own. First, the choice, that the topology of network
N be the one with the smallest number of anonymous nodes, is an arbitrary
one. Second, it turns out that the problem of computing the network topology
with the smallest number of anonymous nodes, from a given set of traces, is
NP-complete. In order to solve this second problem, Jin et al. [9] and Gunes
et al.[7] propose several heuristics (with complexity polynomial in the number
of unique identifiers) that can be used to compute a network topology with a
“small” number of anonymous nodes. Clearly, these heuristics cannot always
compute a network topology with the smallest possible number of anonymous
nodes.
In this paper, we take a different approach to the problem of computing one
unique topology for network N from a given trace set T that is generated by
executing Traceroute over N . Our approach is based on the assumption that the
given trace set T satisfies a number of “conditions”. The assumption, that the
given trace set T satisfies these conditions, is made with the hope that the computed topology for network N is unique. These conditions can be summarized
as follows: (Formal statements of these conditions are given in section 2).
– Unique node identifiers: Each node in network N has exactly one unique
identifier, and if this node occurs in a trace in T , then it occurs in this trace
either by this unique identifier, or by an anonymous identifier.
– Complete coverage: Each edge in network N occurs in at least one trace in
the trace set T . Also, each node in N occurs by its unique identifier in at
least one trace in T .
– Stable and symmetric routing: The routing tables in the nodes of network N
indicate exactly one route between any two nodes in N .
These conditions may appear to be too strong to hold in practice. However,
it is straightforward to show that if the given trace set T does not satisfy any
one of these conditions, then more than one candidate topology for network N
can be computed from T .
For example, assume that the given trace set T does not satisfy the first
condition: a node occurs by identifier a in one trace in T , and occurs by a
second identifier b in another trace in T . In this case, one can infer at least two
candidate topologies for network N : in one topology, a and b indicate one node
in N , and in the other, they indicate two distinct nodes in N .
Our main result in this paper is negative. This means that, even when the
given trace set T satisfies the above (admittedly strong) conditions, it is not
always possible to compute a unique topology for network N from T . Thus,
adopting the above conditions has the effect of strengthening our primary (negative) results.
2
Network Tracing
A network N is a connected, undirected graph where nodes have unique identifiers. Every node in a network is designated either terminal or non-terminal.
Also, every node is either regular or irregular.
A trace t is generable from a network N iff t is a sequence of node identifiers
that represents a simple path between two terminal nodes in N . A regular node
occurs in t by its unique identifier. An irregular node occurs in t either by its
unique identifier, or by the anonymous identifier ∗i , where i is a unique integer
in t. The first and last nodes of t occur by their unique identifiers in t.
We adopt the following notation in our graphical representations.
1.
2.
3.
4.
A terminal node is represented by a box.
A non-terminal node is represented by a circle.
A regular node x is labeled by its unique identifier “x”.
An irregular node x is labeled “x/*”.
Note that a trace t that is generable from a network N is a sequence of nodes
that corresponds to a simple path in N . Thus, there are two ways to write the
sequence of nodes in t. For example, t can be written as (e, ∗1 , ∗2 , ∗3 , a), or it
can be written as (a, ∗3 , ∗2 , ∗1 , e). We regard the differences between these two
ways of writing t as immaterial. Later on, when we mention that a trace is of
the form (x, . . . , y), we mean that this trace could also be of the form (y, . . . , x).
For trace t, we adopt the notation |t| to indicate the number of edges in t.
For example, |(e, ∗1 , ∗2 , ∗3 , a)| = 4.
A trace set T is generable from a network N iff T satisfies the following five
conditions :
1.
2.
3.
4.
5.
T is a set of traces, each of which is generable from N .
For every pair of terminal nodes x, y in N , T has at least one trace (x, . . . , y).
Every edge in N occurs in at least one trace in T .
The unique identifier of every node in N occurs in at least one trace in T .
T is consistent: for every two distinct nodes x and y, if x and y occur in two
or more traces in T , then the exact same set of nodes occur between x and
y in every trace in T where both x and y occur.
Two comments concerning Condition 5 in this definition are in order. First,
if a trace set T has two traces of the form (x, ∗2 , z) and (u, x, y, z), then from
Condition 5, we can conclude that node ∗2 is in fact node y.
Second, if a trace set T has a trace of the form (x, ∗2 , z), then from Condition
5, T cannot have a trace of the form (u, x, ∗5 , y, z). This is because the number
of nodes between x and z in the first trace is 1, and their number in the second
trace is 2, in violation of Condition 5.
The network tracing problem is to design an algorithm that takes as input a
trace set T that is generable from a network, and produces a network N such
that T is generable from N and not from any other network.
3
Impossibility of Network Tracing
Obviously, the network tracing problem is solvable for regular networks, those
where every node is regular. However, it turns out that the problem is not solvable for general networks. In fact, if a network is permitted to have just one
irregular node, then the network tracing problem is unsolvable, as shown by the
following theorem.
Theorem 1 There is no algorithm that takes as an input a trace set T that
is generable from a network with one irregular node, and produces as output a
network N with one irregular node such that:
– T is generable from N , and
– T is not generable from any other network with at least one irregular node.
Proof. (By contradiction) Assume that such an algorithm exists. The following
trace set T1 is generable from network N1 in Figure 1.
T1 = {(a, b), (a, ∗1 , d), (a, f ),
(b, c, d), (b, f ),
(d, e, f )}
If T1 is given as an input to the assumed algorithm, the algorithm produces
network N1 as output. This implies that T1 is not generable from any other
network, which contradicts the fact that T1 is also generable from network N2
in Figure 1.
t
u
Theorem 1 provides a strong negative result for the network tracing problem.
Nevertheless, we identify, in the next three sections, classes of networks for which
the network tracing problem is solvable:
1. Tree networks in Section 4.
2. Odd rings in Section 5.
3. Mostly regular even rings in Section 6.
4
Tracing of Tree Networks
A network N is called a tree iff N is acyclic. In this section, we show that the
network tracing problem is solvable for tree networks.
Theorem 2 There is an algorithm that takes as an input a trace set T that is
generable from a tree network, and produces as output a tree network N such
that:
– T is generable from N , and
– T is not generable from any other tree network.
Fig. 1. Theorem 1
(a) Network N1
(b) Network N2
Proof. (By construction) We prove Theorem 2 by providing the algorithm mentioned. The algorithm consists of the following eight steps:
1. Initially, tree N is empty.
2. Apply procedure Leaf, discussed below, to compute, from T , the unique
identifier of each leaf node in N .
3. Apply procedure Parent, discussed below, to compute from T , the unique
identifier of the parent of each leaf node in N .
4. For every node y that is the parent of a leaf node x, add to tree N an
(undirected) edge between nodes x and y.
5. For every node y that is the parent of a leaf node x, replace in T each trace
of the form (x, ∗i , . . .) by a trace of the form (x, y, . . .).
6. Shorten the traces in T by replacing in T each trace of the form (x, y, . . .),
where x is a leaf node, by the trace (y, . . .) and by discarding from T each
trace that has only one node or is empty.
7. Repeat the algorithm, starting from Step 2, on the trace set T , that results
from Step 6, provided that the resulting set T is non-empty.
8. The algorithm outputs N and terminates when the resulting T from Step 6
is empty.
Next, we specify the two procedures Leaf and Parent that are used in Steps
2 and 3, respectively, of the above algorithm. The correctness of procedure Leaf
follows from the observation that each leaf node in N occurs as a terminal node
in some trace in T , but the converse is not necessarily true. Procedure Leaf is
specified as follows:
procedure Leaf
for each terminal node y in any trace in T
if T has three traces
t = (x, . . . , y), t0 = (y, . . . , z), t00 = (x, . . . , z),
such that |t| + |t0 | = |t00 |
then y is a non-leaf node in N
else y is a leaf node in N
end
The correctness of procedure Parent follows from the observation that the
parent of each leaf node in N occurs by its unique identifier in some trace in T .
Procedure Parent is specified as follows:
procedure Parent
for each leaf node x in N,
if T has a trace of the form (x, y . . .),
or T has two traces of the form (x, ∗i , z) and (z, y, . . .)
where z is a leaf node in N
then the unique identifier of the parent of node x is y
end
t
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5
Tracing of Ring Networks
In this section, we discuss the solvability of the network tracing problem for ring
networks. Surprisingly, we show that the problem is solvable for odd rings (i.e.
cycles with an odd number of nodes), but not solvable for even rings (i.e. cycles
with an even number of nodes).
Theorem 3 There is an algorithm that takes as an input a trace set T that is
generable from an odd ring network, and produces as output an odd ring network
N such that:
– T is generable from N , and
– T is not generable from any other odd ring network.
Proof. (By construction) We prove Theorem 3 by describing the algorithm that
is mentioned in the theorem. The algorithm consists of the following five steps:
1. Construct an unlabeled ring N with n nodes, where n is the number of
unique identifiers that occur in the traces in T . The algorithm terminates
when each node in N is labeled by a distinct unique identifier from those
that occur in the traces in T .
2. Choose any trace t = (a, . . . , b) in T . Label any node in N with the unique
identifier “a”, and label the node in N , that is reachable by traversing |t|
edges clockwise starting from node a, with the unique identifier “b”.
3. For every pair of traces t0 = (a, . . . , c) and t00 = (b, . . . , c) in T ,
if |t| = |t0 | + |t00 | or |t0 | = |t| + |t00 |
then label the node in N , that is reachable by traversing |t0 | edges clockwise
starting from node a, with the unique identifier “c”.
else label the node in N , that is reachable by traversing |t0 | edges counterclockwise starting from node a, with the unique identifier “c”.
4. Note that by the end of Step 3, every unique identifier of a terminal node in
a trace in T is used to label one node in ring N .
5. Consider any trace t0 = (x, . . . , y) in T , and note that |t0 | cannot be equal
to n/2 since |t0 | is a positive integer, and n is odd. Consequently, one can
determine whether any trace t = (x, . . . , y) goes, either clockwise or counterclockwise, from node x to node y. Thus, if trace t has a unique identifier z
that has not yet been used to label any node in N , then one can identify the
node in N that should be labeled with z.
t
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Theorem 4 There is no algorithm that takes as an input a trace set T that
is generable from an even ring network, and produces as output an even ring
network N such that:
– T is generable from N , and
– T is not generable from any other even ring network.
Proof. (By contradiction) The proof proceeds as for Theorem 1, using the observation that the following trace set T2 is generable from two even ring networks,
N3 and N4 , in Figure 2.
T2 = {(a, 1, ∗1 , 6, ∗2 , ∗3 , b), (a, ∗4 , c), (a, ∗5 , ∗6 , ∗7 , d), (a, ∗8 , e), (a, ∗9 , ∗10 , ∗11 , f ),
(b, ∗12 , ∗13 , ∗14 , c), (b, ∗15 , d), (b, ∗16 , ∗17 , ∗18 , e), (b, ∗19 , f ),
(c, ∗20 , ∗21 , 2, ∗22 , 3, d), (c, ∗23 , ∗24 , ∗25 , e), (c, ∗26 , f ),
(d, ∗27 , e), (d, ∗28 , ∗29 , ∗30 , f ),
(e, ∗31 , ∗32 , 4, ∗33 , 5, f )}
t
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Fig. 2. Theorem 5
(a) Network N3
(b) Network N4
6
Tracing of Mostly-regular Networks
A network, where each node has at most one irregular neighbor, is called mostlyregular. The following theorem shows that the network tracing problem is solvable for mostly-regular even rings.
Theorem 5 There is an algorithm that takes as an input a trace set T that
is generable from a mostly-regular even ring network, and produces as output a
mostly-regular even ring network N such that:
– T is generable from N , and
– T is not generable from any other mostly-regular even ring network.
Proof. (By construction). The algorithm is given in our technical report [1]. t
u
Encouraged by Theorem 5, one may have hoped that the network tracing
problem is solvable for the whole class of mostly-regular networks. Unfortunately,
as shown by the next theorem, this turns out not to be the case.
Theorem 6 There is no algorithm that takes as an input a trace set T that
is generable from any mostly-regular network, and produces as output a mostlyregular network N such that:
– T is generable from N , and
– T is not generable from any other mostly-regular network.
Proof. (By contradiction) Suppose such an algorithm exists. As any network
with exactly one irregular node is clearly mostly-regular, this algorithm takes
any trace set generable from a network N with one irregular node, and returns
N (and only N ). This contradicts Theorem 1.
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7
The Weak Network Tracing Problem
The reason that the network tracing problem is not solvable in most cases, one
may argue, is that the given trace set T is required to be generable from one,
and only one, network N . One may hope, then, that if this strict requirement
is somewhat relaxed, then the resulting weak version of the network tracing
problem becomes solvable in many cases. The weak network tracing problem can
be stated as follows:
”Design an algorithm that takes as input a trace set T , that is generable
from a network, and produces a small set {N1 , .., Nk } of networks such that T is
generable from each network in this set and not from any network outside this
set.”
The requirement that the produced set {N1 , .., Nk } be small means, mathematically, that the cardinality k of this set be a constant rather than a function
of the number of unique node identifiers in the given trace set T .
There are both practical and theoretical reasons for this requirement. From a
practical point of view, the smaller the produced set, the better. From a theoretical point of view, allowing the cardinality of the produced set to be a function
of n, the number of unique identifiers, makes the weak network tracing problem trivially solvable (by exhaustive enumeration, setting each ∗i to each unique
identifier).
Unfortunately, the following theorem shows that the weak network tracing
problem is not solvable in general.
Theorem 7 There is no algorithm that takes as an input a trace set T that is
generable from a network, and computes a set of networks {N1 . . . Nk } such that:
– T is generable from every network in the set {N1 . . . Nk },
– T is not generable from any other network, and
– k is a constant whose value is not a function of the number of node identifiers
in T .
Proof. We prove this theorem by exhibiting an infinite sequence of trace sets
T S6 , T S8 , T S10 . . . such that each trace set T Sn satisfies the following three
conditions:
– T Sn has 1 anonymous node identifier.
– T Sn has n unique node identifiers.
– T Sn is generable from any one of n−2
2 distinct networks.
Consider the first trace set in the sequence.
T S6 = {(a, x1 , b), (a, ∗1 , m1 ), (a, x2 , m2 ),
(b, ∗2 , m1 ), (b, x2 , m2 ),
(m1 , m2 )}
This trace set is generable from the two networks N5 and N6 in Figure 3.
We now add two nodes, x3 and m3 , to the trace set T S6 to form the trace
set T S8 and increase the number of possible networks (from which the trace set
T S8 is generable) by 1. Node x3 connects m3 to both a and b, so we add the
traces (a, x3 , m3 ) and (b, x3 , m3 ). Also, m3 is directly connected to every mi , so
we add (m1 , m3 ) and (m2 , m3 ). The resulting trace set T S8 is as follows:
T S8 = {(a, x1 , b), (a, ∗1 , m1 ), (a, x2 , m2 ), (a, x3 , m3 ),
(b, ∗2 , m1 ), (b, x2 , m2 ), (b, x3 , m3 ),
(m1 , m2 ), (m1 , m3 ), (m2 , m3 )}
The trace set T S8 is generable from any of the three networks N7 , N8 and N9
in Figure 4.
By repeating this procedure k − 2 times, we produce a trace set that is
generable from every member of a set of k distinct networks. As each step adds
two new nodes, each of these networks has 2k + 2 nodes. The number of unique
t
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identifiers being n, we have k = n−2
2 .
Fig. 3. Two networks.
(a) Network N5
8
(b) Network N6
Discussion and Related Work
There have been three attacks on the anonymity problem. Casting it as an optimization problem, Yao et al. [13] try building the smallest possible topology
by combining anonymous nodes. They consider two constraints, trace preservation and distance preservation. Proving that optimum topology inference under
these conditions is NP-complete, they propose a O(n5 ) heuristic that merges
anonymous nodes, keeping the constraints invariant. Distance preservation requires that merging nodes never reduces the length of the shortest path between
any two nodes in the computed network; this assumes not only consistency, but
also shortest-path routing. Their study also assumes that anonymous nodes are
strictly anonymous, and their unique identifiers are never revealed.
Jin et al. propose two heuristics to address the problem in [9]. The first
is O(n3 ), uses link delays or node connectivity as attributes, and performs
ISOMAP-based dimensionality reduction. It ignores the difficulty of estimating one-hop delays from RTTs in path traces [3]. The second, a simple O(n2 )
neighbor matching heuristic, has high rates of both false positives and false negatives. In [7], Gunes et al. apply five heuristics in succession and get performance
strictly better than O(n3 ).
This paper addresses the problem and provides a theoretical basis for stating
which instances of trace set can be used to compute exactly one network, and
which cannot. We also give polynomial-time exact algorithms for several network
cases of interest.
Fig. 4. Three networks
(a) Network N7
(b) Network N8
(c) Network N9
9
Concluding Remarks
We have made three contributions in this paper. First, we formally state the
network tracing problem. We then develop the theory by identifying some important network classes for which this problem is solvable, and some for which
it is not. This includes some very surprising results.
We then extend this research using a weaker version of the network tracing
problem, and show that it is not only not possible in general to take a trace set
T and compute a single network N , such that T is generable from N and only
N , it is also not possible to generate a small (with constant cardinality) set of
networks such that T is generable only from members of this set.
In future work, we intend to investigate whether it is possible to relax our
assumptions (consistent routing, unique identifiers, and complete coverage) while
maintaining the effectiveness and elegance of the theory.
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